AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
, 
, 
, 
, 
, 
initially contains one coin. The following operations are allowed
, 
, remove one coin from and add two coins to 
;
, 
, remove one coin from and swap the contents (maybe empty) of the boxes 
and 
., , , , become empty, while box contains exactly 
coins.
+1 w
be a real number and let 
be arbitrary real numbers in 
. The sequences 
and 
are then defined recursively by
. Prove that 
for all sufficiently large 
.
is a prime number
is inscribed in circle 
. Let 
denote its incenter and 
its 
-excenter. Let 
denote the midpoint of arc 
. Line 
meets a second time at 
. The perpendicular to 
at meets sides 
and 
at 
and 
respectively. The circumcircle of 
meets 
a second time at 
, and the circumcircle of 
meets 
a second time at 
. Prove that 
passes through the antipodal to on .
be a triangle with circumcircle and incentre . A line 
intersects the lines , 
, and 
at points 
, , and , respectively, distinct from the points , 
, 
, and . The perpendicular bisectors 
, 
, and 
of the segments 
, 
, and 
, respectively determine a triangle 
. Show that the circumcircle of the triangle is tangent to .
be a positive integer. Prove that there are infinitely many primes, each of which divides at least one term of the sequence 
, 
.
![$\mathbb{Z}[x]$](http://latex.artofproblemsolving.com/4/b/7/4b7d768fffbb7a1a6163af392ed348f2d49d8783.png)
denote the set of single-variable polynomials in with integer coefficients. Find all functions ![$\theta : \mathbb{Z}[x] \to \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/0/3/2/032a637d4ae21da2c0ddd8dc127723807f11cc5f.png)
(i.e. functions taking polynomials to polynomials)![$p, q \in \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/7/e/7/7e7eb925258a33e456ef4ff722c1f943dc155d10.png)
, 
;![$p \in \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/0/2/6/0267b98c16b400dda1f081877e9b84fe672ce0e7.png)
, 
has an integer root if and only if 
does.
of positive real numbers that satisfy the system:![\[
\begin{aligned}
11bc - 36b - 15c &= abc \\
12ca - 10c - 28a &= abc \\
13ab - 21a - 6b &= abc.
\end{aligned}
\]](http://latex.artofproblemsolving.com/1/7/0/170e1cfaee90f5c37428fa75192ac889261a8f83.png)
that verify the following equation for all 
:
is a function for which the expression 
for all 
is always a perfect square. Prove that 
for all 
.
such that 
.
be a quadrilateral inscribed in a circle 
, and let be a point on the extension of such that 
and 
are tangent to . The tangent at intersects at and the line at 
. Let be the second point of intersection between 
and . Prove that , , are collinear.
be a triangle with incenter , and suppose the incircle is tangent to 
and at and . Denote by 
and 
the reflections of and over . Let be the intersection of 
with 
, and let 
be the midpoint of . Prove that 
and 
are perpendicular.
be an acute scalene triangle with circumcenter 
. Let 
be the feet of the altitudes from 
to 
respectively. Let be the midpoint of . Let 
be the circle with diameter . Let 
be the intersection of 
and . Let be the orthocenter of . Let 
be the intersection of and . Let 
be the lines through tangent to 
respectively. Let be the intersection of 
and 
. Let be the intersection of 
and 
. Let 
be the line through parallel to 
and let 
be the reflection of across . Prove that 
is tangent to 
.
is a midpoint of side of a triangle and is the orthocenter of . 
intersects the -angle bisector at . Points 
and 
are the projections of on sides and . Prove that 
passes through .
is concyclic and 
, as desired. Now, by Fact 5, we get .
, 
and 
be reflections of , and about , its well-known that lies on and 
is the diameter of , hence 
. To prove , it suffices to show that 
. Now, just note that 
, as desired we get 
. Hence, 
. Now, just note that 
. Hence, 
. Thus, 
are collinear , as desired 
at 
respectively.
.
and by Simson Line thm 
.